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Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S -primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S -primary ideal if there exists s ∈ S such that whenever 0 ≠ a b ∈ I for a , b ∈ R , then s a ∈ I or s b ∈ I. The relationships among S -prime, S -primary, weakly S -primary and S - n -ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S r -primary ideals are characterized where S r = { 1 , r , r 2 , ... }. Several properties, characterizations and examples concerning weakly S -primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S -decomposable ideals and S -weakly Laskerian rings which are generalizations of S -decomposable ideals and S -Laskerian rings are introduced. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative ring with identity. Let φ : S(R) -- S(R) → { Φ} be a function where S(R) denotes the set of all subsets of R. The aim of this paper is to generalize the concept of prime radical √I of an ideal I of R to φ-prime radical Pφ (I). A proper ideal Q of R is called φ-prime if whenever a, b ∈ R, aRb ⊆ Q and aRb ... φ (Q) implies that either a ∈ Q or b ∈ Q. In this paper, first we study the properties of several generalizations of prime ideals of R. Then, we verify that P φ(I) is equal to the intersection of all minimal φ-prime ideals of R containing I, and we show that this notion inherits many of the essential properties of the usual notion of prime radical of an ideal. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. For positive integers m and n, Anderson and Badawi (Journal of Algebra and Its Applications 16(1):1750013 (21 pp), 2017) defined an ideal I of a ring R to be an (m,n)-closed if whenever x m ∈ I , then x n ∈ I . In this paper we define and study a new generalization of the class of (m,n)-closed ideals which is the class of quasi (m,n)-closed ideals. A proper ideal I is called quasi (m,n)-closed in R if for x ∈ R , x m ∈ I implies either x n ∈ I or x m - n ∈ I . That is, I is quasi (m,n)-closed in R if and only if I is either (m, n)-closed or ( m , m - n )-closed in R. We justify several properties and characterizations of quasi (m,n)-closed ideals with many supporting examples. Furthermore, we investigate quasi (m,n)-closed ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behavior of this class of ideals in idealization rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and let S be a multiplicative subset of R. Then R is called a nonnil-S-Noetherian ring if every nonnil ideal of R is S-finite. Also, R is called a u-S-Noetherian ring if there exists an element s ∈ S such that for each ideal I of R, sI ⊆ K for some finitely generated sub-ideal K of I. In this paper, we examine some new characterization of nonnil-S-Noetherian rings. Then, as a generalization of nonnil-S-Noetherian rings and u-S-Noetherian rings, we introduce and investigate the nonnilu-S-Noetherian rings class. [ABSTRACT FROM AUTHOR]

For commutative rings with identity, we introduce and study the concept of semi r-ideals which is a kind of generalization of both r-ideals and semiprime ideals. A proper ideal I of a commutative ring R is called semi r-ideal if whenever a² ∈ I and AnnR(a) = 0, then a ∈ I. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi r-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi r-ideals of rings to semi r-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the D-annihilator condition and justify when they are semi r-submodules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N:R M) or m ∈ M - rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal I of R is called a 1 -absorbing primary ideal of R if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ I. Some properties of 1-absorbing primary ideals are investigated. For example, we show that if R admits a 1-absorbing primary ideal that is not a primary ideal, then R is a quasilocal ring. We give an example of a 1-absorbing primary ideal of R that is not a primary ideal of R. We show that if R is a Noetherian domain, then R is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of R is of the form P n for some nonzero prime ideal P of R and a positive integer n ≥ 1. We show that a proper ideal I of R is a 1-absorbing primary ideal of R if and only if whenever I 1 I 2 I 3 ⊆ I for some proper ideals I 1 , I 2 , I 3 of R , then I 1 I 2 ⊆ I or I 3 ⊆ I. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1 ≠ 0 and M an R-module. We will call a proper submodule N of M as a semi n-absorbing submodule of M if whenever r∊R, m∊M with rnm∊N, then rn∊(N:R M) or rn-1m∊N. We will say N to be a (k, n)-closed submodule of M if whenever r∊R, m∊M with rkm∊N, then rn∊(N:R M) or rn-1m∊N. In this paper we introduce semi n-absorbing and (k, n)-closed submodules of modules over commutative rings, and investigate their basic properties. [ABSTRACT FROM AUTHOR]

A proper element q of a lattice L is said to be a quasi n-absorbing element if whenever anb ⩽ q implies that either an ⩽ q or an-1b ⩽ q. We investigate properties of this new type of elements and obtain some relations among prime, 2-absorbing, n-absorbing elements in multiplicative lattices. [ABSTRACT FROM AUTHOR]